[FOM] Banach Tarski Paradox/Line

[Christian Espindola]

The paradox cannot take place in R or in R^2, so the answer would be no.
This follows from the existence of a Banach measure in both R and R^2,
i.e., a finitely additive positive measure that is invariant under rigid
transformations.

The fact that dimensions 1 and 2 allow the existence of Banach measures,
while it cannot exist in dimension 3, is due in turn to the fact that the
group G of rigid transformations of R^3 is not soluble, while the
corresponding groups in R and R^2 are indeed soluble. The construction of a
Banach measure is performed with the aid of a G - invariant version of
Hahn-Banach theorem, which needs G to be soluble to apply. Actually, it is
the very fact that in R^3 the structure of the group G is much more complex
and rich what makes the paradox argument work. Note, in fact, that in R^3,
G is not soluble since it contains the nonabelian free group in two
generators as a subgroup, which is precisely the subgroup that helps
constructing paradoxical decompositions like the one in Banach-Tarski
paradox.

Christian.

On Sun, Nov 27, 2011 at 6:27 PM, <pax0 at seznam.cz> wrote:

> Is the Banach Tarski paradox provable for the unit real interval;
> i.e. is there a possibility for duplicating [0,1].
> If not, where is the obstacle?
> Jan Pax
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